A kinetic neutral atom model for tokamak scrape-off layer tubulence simulations. Christoph Wersal, Paolo Ricci, Federico Halpern, Fabio Riva

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1 A kinetic neutral atom model for tokamak scrape-off layer tubulence simulations Christoph Wersal, Paolo Ricci, Federico Halpern, Fabio Riva CRPP - EPFL SPS Annual Meeting CRPP

2 The tokamak scrape-off layer (SOL) Open field lines Heat exhaust Confinement Impurities Fusion ashes removal Fueling the plasma (recycling) Three regimes Convection limited Conduction limited Detached Christoph Wersal - CRPP Neutral particles and SOL turbulence 2 / 14

3 Convection limited regime Christoph Wersal - CRPP Neutral particles and SOL turbulence 3 / 14

4 Convection limited regime Christoph Wersal - CRPP Neutral particles and SOL turbulence 3 / 14

5 Convection limited regime Christoph Wersal - CRPP Neutral particles and SOL turbulence 3 / 14

6 Convection limited regime Low plasma density Long λ mfp for neutrals Neutrals Christoph Wersal - CRPP Neutral particles and SOL turbulence 3 / 14

7 Convection limited regime Low plasma density Long λ mfp for neutrals Ionization in the core Ionization Neutrals Christoph Wersal - CRPP Neutral particles and SOL turbulence 3 / 14

8 Convection limited regime Low plasma density Long λ mfp for neutrals Ionization in the core Heat particle source Ionization Neutrals Christoph Wersal - CRPP Neutral particles and SOL turbulence 3 / 14

9 Convection limited regime Low plasma density Long λ mfp for neutrals Ionization in the core Heat particle source Ionization Q is mainly convective Neutrals Christoph Wersal - CRPP Neutral particles and SOL turbulence 3 / 14

10 Conduction limited regime High plasma density Christoph Wersal - CRPP Neutral particles and SOL turbulence 4 / 14

11 Conduction limited regime High plasma density Short λ mfp Neutrals Christoph Wersal - CRPP Neutral particles and SOL turbulence 4 / 14

12 Conduction limited regime High plasma density Short λ mfp Ionization close to targets Ionization Neutrals Christoph Wersal - CRPP Neutral particles and SOL turbulence 4 / 14

13 Conduction limited regime High plasma density Short λ mfp Ionization close to targets Temperature gradients form Q is mainly conductive Neutrals Ionization Christoph Wersal - CRPP Neutral particles and SOL turbulence 4 / 14

14 The GBS code, a tool to simulate SOL turbulence Drift-reduced Braginskii equations d/dt ω ci,k 2 k 2 Evolves scalar fields in 3D geometry n,ω,v e,v,i,t e,t i Flux-driven, no separation between equilibrium and fluctuations Power balance between plasma outflow from the core, turbulent transport, and sheath losses [Ricci et al., PPCF, 2012] No divertor geometry No neutral physics Christoph Wersal - CRPP Neutral particles and SOL turbulence 5 / 14

15 The model Introduction The neutrals model Conclusions Kinetic model for the neutral atoms with Krook operators f n t + v f n x = ν ionf n ν cx (f n n n Φ i ) + ν rec n i Φ i (1) ν ion = n e v e σ ion (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Boundary conditions Φ i = f i /n i (v in respect to the surface; θ between v and normal vector to the surface) dv v f n ( x w, v) + u i n i = 0 (2) f n ( x w, v) cos(θ)e mv 2 /2T w for v > 0 Christoph Wersal - CRPP Neutral particles and SOL turbulence 6 / 14

16 The model Introduction The neutrals model Conclusions Kinetic model for the neutral atoms with Krook operators f n t + v f n x = ν ionf n ν cx (f n n n Φ i ) + ν rec n i Φ i (1) ν ion = n e v e σ ion (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Boundary conditions Φ i = f i /n i (v in respect to the surface; θ between v and normal vector to the surface) dv v f n ( x w, v) + u i n i = 0 (2) f n ( x w, v) cos(θ)e mv 2 /2T w for v > 0 Christoph Wersal - CRPP Neutral particles and SOL turbulence 6 / 14

17 The model Introduction The neutrals model Conclusions Kinetic model for the neutral atoms with Krook operators f n t + v f n x = ν ionf n ν cx (f n n n Φ i ) + ν rec n i Φ i (1) ν ion = n e v e σ ion (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Boundary conditions Φ i = f i /n i (v in respect to the surface; θ between v and normal vector to the surface) dv v f n ( x w, v) + u i n i = 0 (2) f n ( x w, v) cos(θ)e mv 2 /2T w for v > 0 Christoph Wersal - CRPP Neutral particles and SOL turbulence 6 / 14

18 The model Introduction The neutrals model Conclusions Kinetic model for the neutral atoms with Krook operators f n t + v f n x = ν ionf n ν cx (f n n n Φ i ) + ν rec n i Φ i (1) ν ion = n e v e σ ion (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Boundary conditions Φ i = f i /n i (v in respect to the surface; θ between v and normal vector to the surface) dv v f n ( x w, v) + u i n i = 0 (2) f n ( x w, v) cos(θ)e mv 2 /2T w for v > 0 Christoph Wersal - CRPP Neutral particles and SOL turbulence 6 / 14

19 The model Introduction The neutrals model Conclusions Kinetic model for the neutral atoms with Krook operators f n t + v f n x = ν ionf n ν cx (f n n n Φ i ) + ν rec n i Φ i (1) ν ion = n e v e σ ion (v e ), ν cx = n i v rel σ cx (v rel ) ν rec = n e v e σ rec (v e ), Boundary conditions Φ i = f i /n i (v in respect to the surface; θ between v and normal vector to the surface) dv v f n ( x w, v) + u i n i = 0 (2) f n ( x w, v) cos(θ)e mv 2 /2T w for v > 0 Christoph Wersal - CRPP Neutral particles and SOL turbulence 6 / 14

20 The model in steady state Steady state, f n t = 0, first approach Valid if τ neutral losses < τ turbulence e.g. T e = 20eV, n 0 = m 3 τ loss ν 1 eff s τ turbulence R 0 L p /c s s Otherwise: time dependent model Christoph Wersal - CRPP Neutral particles and SOL turbulence 7 / 14

21 The method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v 0 x Christoph Wersal - CRPP Neutral particles and SOL turbulence 8 / 14

22 The method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v 0 x f n (x,v) (3) Christoph Wersal - CRPP Neutral particles and SOL turbulence 8 / 14

23 The method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 v 0 x f n (x,v) = x 0 dx (3) Christoph Wersal - CRPP Neutral particles and SOL turbulence 8 / 14

24 The method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) v (3) Christoph Wersal - CRPP Neutral particles and SOL turbulence 8 / 14

25 The method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) e 1 x v x dx (ν cx (x )+ν ion (x )) v (3) Christoph Wersal - CRPP Neutral particles and SOL turbulence 8 / 14

26 The method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) e 1 x v x dx (ν cx (x )+ν ion (x )) v (3) + f w (v)e 1 v x0 dx (ν cx (x )+ν ion (x )) Christoph Wersal - CRPP Neutral particles and SOL turbulence 8 / 14

27 The method of characteristics Example in 1D, no recombination, v > 0 and a wall at x = 0 0 x' x v f n (x,v) = x 0 dx ν cx (x )n n (x )Φ i (x,v) e 1 x v x dx (ν cx (x )+ν ion (x )) v (3) + f w (v)e 1 v x0 dx (ν cx (x )+ν ion (x )) Christoph Wersal - CRPP Neutral particles and SOL turbulence 8 / 14

28 An equation for the density distribution By imposing dv f n = n n n n (x) = x dx n n (x ) contribution by v < 0 + n w (x) dv ν cx(x )Φ i (x,v) v e ν eff (x x ) v (4) we get an integral equation for n n (x). Christoph Wersal - CRPP Neutral particles and SOL turbulence 9 / 14

29 Discretized equations Discretize the spatial direction in x x i and n n (x i ) n i n nn i = nn j j i 0 dv ν cx(x j )Φ i (x j,v) e ν eff (x i x j ) v (5) v + contribution by v < 0 + n w (x i ) System of linear equations, solved with standard methods Christoph Wersal - CRPP Neutral particles and SOL turbulence 10 / 14

30 Self-consistent simulation with GBS and neutrals model Plasma: snapshot of density, parallel ion velocity and electron temperature from GBS Christoph Wersal - CRPP Neutral particles and SOL turbulence 11 / 14

31 Self-consistent simulation with GBS and neutrals model Neutral density distribution and ionization rate (n n n e v e σ ion ) from the simple neutral model, n 0 = m 3, T 0 = 10eV Christoph Wersal - CRPP Neutral particles and SOL turbulence 12 / 14

32 Conclusions Developement of a neutral model for GBS Kinetic equation with Krook operators for ion, rec and cx 2D evaluation of the neutral density shows its proximity to the limiter Christoph Wersal - CRPP Neutral particles and SOL turbulence 13 / 14

33 Outlook Find the transition from convection to conduction limited regime Relax the steady state assumption Adding density source, drag force, and temperature source/sink to the equations in GBS Move towards detachment Divertor geometry Radiative detachment in limited geometry Mimic the geometry of one divertor leg Christoph Wersal - CRPP Neutral particles and SOL turbulence 14 / 14

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